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Find the volume of the solid bounded by the graphs of the equations

z = x + y, x² + y² = 4 in the first octant.

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z = x + y, x² + y² = 4 in the first octant.

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Visualize the solid. It is like an inverted cone. But the cone angle is not a constant. The outer periphery of the solid (base of the cone like solid) is given by x² + y² = 2². or, r² = 2²

polar coordinates:

x= r cosФ and y = r sinФ

dA in a z plane : (dr) (r dФ)

The limits are 0<= r <= 2 and 0<= Ф <= π/2 for the first octant.

dV = z dA = (x + y) dr r dФ = r² (cosФ + sinФ) dr dФ

Volume

so the volume is 16/3

polar coordinates:

x= r cosФ and y = r sinФ

dA in a z plane : (dr) (r dФ)

The limits are 0<= r <= 2 and 0<= Ф <= π/2 for the first octant.

dV = z dA = (x + y) dr r dФ = r² (cosФ + sinФ) dr dФ

Volume

so the volume is 16/3